To get MENTAL solution, consider a rectangle of area 1 square unit inscribed in a circle of area  square units.


This circle has the radius 1; hence, its diameter is 2 units.


Now your rectangle has the diagonal of the length 2 units and the area of 1 square unit.


Let x and y be the dimensions of the rectangle. Then


    x^2 + y^2 = 2^2 = 4,     (1)   and

    xy              = 1.     (2)


From (1) and (2),


    x^2 + 2xy + y^2 = 4 + 2 = 6;  hence

    (x + y)^2 = 6,


which implies 

    x + y = .


Thus the perimeter of this rectangle is  2*(x+y) = .



Now dilate your rectangle and your circle similarly with the similarity (= dilation) coefficient of   = .


Then you will get exactly the rectangle and the circle of the original condition.


Hence, the perimeter of the rectangle under the question is  

     P =  = .    ANSWER